Assume quantum states A=[1/sqrt(15), -1/sqrt(15), 2/sqrt(15), -3/sqrt(15)] and B=[-1/sqrt(15), -1/sqrt(15), 2/sqrt(15), 3/sqrt(15)]. How does physics or quantum circuits distinguish between quantum A and B?

Hey @R-X , that’s always a good starting question.

If you’d like to learn about this step-by-step, I can recommend the node I.1 in the Quantum Codebook. It will show you exactly how you can figure this out using both physics/mathematics and code!

As a short answer, you can, for example, calculate the inner product of two normalized states (one as a bra, the other as a ket), and see if you get a 1 (same state), a 0 (orthogonal states), or some other number (something in between).

I can really recommend that you take some time checking out these topics in the Codebook — the examples can be really helpful.